Numerical resolution of Emden's equation using Adomian polynomials
International Journal of Numerical Methods for Heat & Fluid Flow
ISSN: 0961-5539
Article publication date: 2 August 2013
Abstract
Purpose
In this paper the authors aim to show the advantages of using the decomposition method introduced by Adomian to solve Emden's equation, a classical non‐linear equation that appears in the study of the thermal behaviour of a spherical cloud and of the gravitational potential of a polytropic fluid at hydrostatic equilibrium.
Design/methodology/approach
In their work, the authors first review Emden's equation and its possible solutions using the Frobenius and power series methods; then, Adomian polynomials are introduced. Afterwards, Emden's equation is solved using Adomian's decomposition method and, finally, they conclude with a comparison of the solution given by Adomian's method with the solution obtained by the other methods, for certain cases where the exact solution is known.
Findings
Solving Emden's equation for n in the interval [0, 5] is very interesting for several scientific applications, such as astronomy. However, the exact solution is known only for n=0, n=1 and n=5. The experiments show that Adomian's method achieves an approximate solution which overlaps with the exact solution when n=0, and that coincides with the Taylor expansion of the exact solutions for n=1 and n=5. As a result, the authors obtained quite satisfactory results from their proposal.
Originality/value
The main classical methods for obtaining approximate solutions of Emden's equation have serious computational drawbacks. The authors make a new, efficient numerical implementation for solving this equation, constructing iteratively the Adomian polynomials, which leads to a solution of Emden's equation that extends the range of variation of parameter n compared to the solutions given by both the Frobenius and the power series methods.
Keywords
Citation
José Pujol, M., Pujol, F.A., Aznar, F., Pujol, M. and Rizo, R. (2013), "Numerical resolution of Emden's equation using Adomian polynomials", International Journal of Numerical Methods for Heat & Fluid Flow, Vol. 23 No. 6, pp. 1012-1022. https://doi.org/10.1108/HFF-05-2011-0109
Publisher
:Emerald Group Publishing Limited
Copyright © 2013, Emerald Group Publishing Limited